Real hypersurfaces with harmonic Weyl tensor of a complex space form
The article is concerned with real hypersurfaces in a complex space form \(M_ n(c)\) of holomorphic sectional curvature \(c\neq 0\) and of dimension \(n\geq 3\), where the real hypersurfaces are assumed to satisfy certain (intrinsic) curvature conditions. The main result is a classification of all complete ``cyclic Ryan'' hypersurfaces of \(M_ n(c)\); its proof is based on the Cecil/Ryan and Montiel's classification of pseudo-Einstein hypersurfaces in \(M_ n(c)\). Moreover, the authors show that there are no real hypersurfaces with harmonic Weyl tensor in \(M_ n(c)\). This assertion generalizes results of Ki, Kim, Kon Kwon/Nakagawa and Montiel; see in particular \textit{H.-J. Kim} [Tsukuba J. Math. 12, No.2, 451-457 (1988; Zbl 0661.53013)] and \textit{J.-H. Kwon} and \textit{H. Nakagawa} [J. Aust. Math. Soc., Ser. A 47, No.1, 108-113 (1989; Zbl 0692.53005)] for the analogous result on hypersurfaces with harmonic curvature.
scientific article; zbMATH DE number 4180241